Intermediate Value Theorem

Pre-Calculus · Axiom Academy

LESSON Intermediate Value Theorem Why a sign flip between two points forces a continuous curve to cross zero — and how that lets us hunt down roots. 1. A Sign Flip Forces a Crossing Suppose a polynomial p is negative at x=a and positive at x=b . Because the graph is one unbroken curve, it cannot jump from below the axis to above it — somewhere between a and b it has to touch down on y=0 . Watch the curve grow from the negative endpoint to the positive one: the moment it changes sign is a root . Opposite signs at the endpoints forces a root c strictly between them 2. Zero Is Trapped Between the Two Values Here is the engine of the theorem, stated for outputs alone. As x travels from a to b , the output p(x) travels continuously from p(a) to p(b) — it hits every value in between. And if p(a) is negative while p(b) is positive, then 0 is one of those in-between values. So p(x)=0 has to happen. The output begins below zero — the value bar starts in the red (negative) region. To reach a positive value from a negative one without skipping, it must land on 0 exactly once on the way up. The output finishes above zero — the bar ends in the green (positive) region. This only works because the output cannot teleport. A graph with a break could leap over 0 . 3. Using the Sign Flip to Locate a Root

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